Algèbres de Hopf des permutahèdres, associahèdres et hypercubes

✍ Scribed by Frédéric Chapoton

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 152 KB
• Volume: 150
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1868

✦ Synopsis

Three graded Hopf algebras over Z are constructed, each one having in degree n a basis indexed by the faces of a polytope of dimension n&1, respectively, the hypercube, the associahedron, and the permutahedron. This lifts to the level of all faces previous constructions of Loday and Ronco and Malvenuto and Reutenauer for vertices of the same polyhedra. 2000 Academic Press 1. ALGE BRE DE HOPF DES PERMUTAHE DRES Soient n 1 et 1 d n. On note Perm(n, d ) l'ensemble des partitions ordonne es de [1, ..., n] en d parts. Il existe un polytope P n de dimension n&1 appele permutaheÁ dre tel que Perm(n, d ) soit en bijection avec les faces de dimension n&d de P n (cf. [3]). On peut aussi de crire Perm(n, d ) comme l'ensemble des applications surjectives de [1, ..., n] dans [1, ..., d].